The derivative describes how a function changes at one exact input value. It is the slope of the tangent line to the graph at a point, not the slope over a long interval. This matters because many quantities in science are changing continuously, such as position, temperature, voltage, or population.
The derivative turns a curved graph into local information about direction and rate of change.
The definition of the derivative starts with the slope of a secant line through two nearby points on a curve. If the points are A = (x, f(x)) and B = (x + h, f(x + h)), the secant slope is [f(x + h) - f(x)] / h. As h gets closer to 0, point B moves toward point A and the secant line approaches the tangent line.
If this limit exists, it gives f'(x), the instantaneous rate of change of f at x.
Understanding Calculus: The Definition of the Derivative
The important idea in the definition is that the input change is never actually set equal to zero during the calculation. If it were zero, the change in the function would be zero too, giving an undefined zero divided by zero result. Instead, the quotient is worked out for nearby nonzero changes.
The limit asks what number those quotients settle toward as the changes become extremely small. This distinction is why a derivative is a limit, not merely a fraction with zero on the bottom.
Algebra often reveals the limit clearly. Consider a function that squares its input. Start at an input called a, then increase it by a small amount called d.
The change in the squared output can be expanded and simplified. After dividing by d, the result becomes two times a plus d. This expression is valid while d is not zero.
As d approaches zero, its extra term disappears, leaving two times a. The cancellation step matters.
Students should simplify first, then take the limit. Plugging in zero before simplifying usually creates the undefined form and hides the answer.
Not every graph has a derivative at every point. A sharp corner gives different nearby slopes when approached from opposite directions. The graph of absolute value has this issue at zero.
Coming from the left, the slopes are negative one. Coming from the right, the slopes are positive one. Since they do not approach one shared number, there is no derivative there.
A vertical tangent can produce slopes that grow without bound. Some functions can even wiggle so rapidly near a point that their slopes never settle.
Continuity is necessary for differentiability, but it is not enough. A graph may be unbroken yet still have a corner.
Derivatives carry units, which helps connect the definition to measurements. If a position is measured in metres and time is measured in seconds, its derivative has units of metres per second. If temperature changes with height, the derivative may be measured in degrees per metre.
Units tell what the rate means in context. Near a point where the derivative exists, the function behaves approximately like a straight line for very small input changes. A derivative of three means that an input increase of about one tenth produces an output increase of about three tenths, provided the interval is small enough.
When learning derivative definitions, pay close attention to the direction of approach. Small positive changes examine the graph from the right. Small negative changes examine it from the left.
Both sides must agree for the usual derivative at an interior point. On a calculator or computer, nearby quotient values can estimate a derivative, but very tiny changes may cause rounding error.
Graphs can be useful evidence, yet algebra and limits provide the proof. Practice by writing the average rate expression, simplifying it carefully, then describing what happens as the input change shrinks.
Key Facts
- Derivative definition: f'(x) = lim h->0 [f(x + h) - f(x)] / h
- Difference quotient: [f(x + h) - f(x)] / h
- Secant slope between x and x + h: msec = [f(x + h) - f(x)] / [(x + h) - x]
- Tangent slope at x: mtan = f'(x), if the limit exists
- Instantaneous velocity from position s(t): v(t) = s'(t) = lim h->0 [s(t + h) - s(t)] / h
- A function is not differentiable at a point if the limiting slopes from the left and right do not agree or become infinite.
Vocabulary
- Derivative
- The derivative f'(x) is the limit of the difference quotient and gives the instantaneous rate of change of f at x.
- Difference quotient
- The difference quotient [f(x + h) - f(x)] / h is the average rate of change over a small interval of width h.
- Secant line
- A secant line is a line that passes through two points on a curve and has slope equal to the average rate of change between them.
- Tangent line
- A tangent line is the limiting position of secant lines as the second point approaches the first point.
- Instantaneous rate of change
- An instantaneous rate of change describes how fast a quantity is changing at one exact input value.
Common Mistakes to Avoid
- Substituting h = 0 too early is wrong because the difference quotient usually becomes 0/0 before simplification. First simplify the expression, then take the limit as h approaches 0.
- Confusing secant slope with tangent slope is wrong because a secant line uses two distinct points while a tangent slope is the limit as those points merge. The derivative is not just any average slope.
- Canceling terms incorrectly across addition is wrong because cancellation only works for common factors. For example, (x^2 + 2xh + h^2 - x^2) / h must be simplified to (2xh + h^2) / h before canceling h as a factor.
- Assuming every continuous graph has a derivative is wrong because corners, cusps, and vertical tangents can prevent a single tangent slope from existing. Continuity is necessary for differentiability, but it is not enough.
Practice Questions
- 1 Use the definition f'(x) = lim h->0 [f(x + h) - f(x)] / h to find f'(x) for f(x) = x^2 + 3x.
- 2 For s(t) = 4t^2 - 2t, compute the instantaneous velocity at t = 3 using the derivative definition or an equivalent limit calculation.
- 3 A graph has a sharp corner at x = 2 where the left-hand slope approaches -1 and the right-hand slope approaches 4. Explain whether the derivative exists at x = 2 and why.